Author: Jens Christian Wahl

# stochvolTMB: Likelihood estimation of stochastic volatility

#### Jens Christian Wahl

# The stochastic volatility model

The stochastic volatility model, introduced by Taylor (1982), is defined by \[\begin{equation} \begin{aligned} y_t &= \sigma_y e^{h_t/2} \epsilon_t, \quad t = 1, \dots, T, \\ h_{t+1} &= \phi h_{t} + \sigma_h \eta_t, \quad t = 1, \dots, T-1, \\ \eta_t &\stackrel{\text{iid}}{\sim} \mathcal{N}(0,1), \\ \epsilon_t &\stackrel{\text{iid}} {\sim} F \end{aligned} \end{equation}\] where \(y_t\) is the observed log returns, \(h_t\) is the logarithm of the variance on day \(t\). The distribution of the innovations \(\epsilon_t\) is specified below. To ensure stationarity for \(h_t\), we assume \(|\phi| < 1\). It can be shown that the unconditional distribution of \(h_t\) is \(\mathcal{N}(0,\sigma_h^2/(1 - \phi^2))\), and we assume \(h_1 \sim \mathcal{N}(0,\sigma_h^2/(1-\phi^2))\). An interpretation of the latent process \(\{h_t\}\) is that is represents the random and uneven flow of new information into the marked. For different time points, the variance will be dependent of this unobserved ``flow’’ of information, i.e. conditioning on \(h_t\), \(\mathrm{Var}(y_t | h_t) = \sigma_x^2 e^{h_t}\).

The original SV model from Taylor (1982) assumed normally distributed innovations, but this is usually to strong of an assumption. Financial returns are usually heavy-tailed, might be asymmetric and the volatility can be correlated with the returns. The latter is called the leverage effect, where there is a negative correlation between a change in price and the volatility, meaning that a drop in price tend to lead to an increase in volatility To take these features of financial time series into account `stochvolTMB` has implemented the following four distribution for the innovations:

- Gaussian: \(\epsilon_t \sim \mathcal{N}(0,1)\)
- t: \(\epsilon_t \sim t_\nu(0,1)\), where \(\nu\) is the degrees of freedom.
- Skew-Gaussian: \(\epsilon_t \sim \mathcal{SN}(0,1, \alpha)\), with zero mean, unit variance and skewness parameter \(\alpha\).
- Leverage: \(\epsilon_t \sim \mathcal{N}(0,1)\) and \(\mathrm{Cor}(\epsilon_t, \eta_t) = \rho\)

`stochvolTMB` is inspired by the package stochvol (Kastner (2016)), but `stochvolTMB` obtain parameter estimates through maximum likelihood estimation and not Markov Chain Monte Carlo.

# Usage

The main functions of `stochvolTMB` are:

Function Name | Description
--- | ---
`estimate_parameters` | Estimate parameters of a stochastic volatility model.
`sim_sv` | Simulate data from a stochastic volatility model.
`plot.stochvolTMB` | Plot estimated volatility and predicted (with confidence intervals).
`summary.stochvolTMB` | Extract parameter estimates and volatility with uncertainty.
`predict.stochvolTMB` | Predict future volatility and returns.

`estimate_parameters` returns an object of class `stochvolTMB`. The `summary` function returns a `data.table` with estimated parameters, estimated log-volatility and transformed parameters along with standard errors, p-values and z-values. The argument `report = "fixed"` returns the parameters on the scale they were estimated. This means that the standard deviations \(\sigma_y, \sigma_h\) are given on the log scale; the degrees of freedom is one the scale \(\log (\nu - 2)\) to ensure that \(\nu > 2\) (i.e. the variance exists); and lastly that the persistence parameter \(\phi\) and the correlation parameter \(\rho\) are estimated on a logit scale to ensure that they are between -1 and 1. If `report = c("fixed", "transformed")`, parameter estimates transformed back to their original scale is also returned. If you want to extract the estimated log-volatility you can use `report = "random"`.

# Example

Estimating parameters in a SV model is easy with `stochvolTMB`. As an example we investigate the daily log-returns of the S&P500 from 2005 to 2018. A quick look at the data:

    data(spy)
    plot(spy$date, spy$log_return, type = "l", xlab = "", ylab = "", main = "Log-returns of S&P500")

    plot(spy$date, spy$price, type = "l", xlab = "", ylab = "", main = "Price of S&P500")

We fit all four distributions:

    gaussian = estimate_parameters(spy$log_return, model = "gaussian", silent = TRUE)
    t_dist = estimate_parameters(spy$log_return, model = "t", silent = TRUE)
    skew_gaussian = estimate_parameters(spy$log_return, model = "skew_gaussian", silent = TRUE)
    leverage = estimate_parameters(spy$log_return, model = "leverage", silent = TRUE)

We can investigate the estimate for the degrees of freedom (`df`) to see if the returns are heavy-tailed

    summary(t_dist, report = "transformed")
    #>    parameter     estimate    std_error    z_value      p_value        type
    #>       <char>        <num>        <num>      <num>        <num>      <char>
    #> 1:   sigma_y  0.008392879 0.0008689248   9.658924 4.505714e-22 transformed
    #> 2:   sigma_h  0.185766767 0.0182175649  10.197124 2.042308e-24 transformed
    #> 3:       phi  0.984924644 0.0039281816 250.732972 0.000000e+00 transformed
    #> 4:        df 10.086366735 2.1022434972   4.797906 1.603330e-06 transformed

Clearly the returns are more heavy tailed than Gaussian, even when controlling for the stochastic volatility. We can also check for asymmetric returns

    summary(skew_gaussian, report = "fixed")
    #>      parameter  estimate  std_error    z_value      p_value   type
    #>         <char>     <num>      <num>      <num>        <num> <char>
    #> 1: log_sigma_y -4.797828 0.09127165 -52.566464 0.000000e+00  fixed
    #> 2: log_sigma_h -1.556355 0.08899553 -17.488014 1.768113e-68  fixed
    #> 3:   logit_phi  4.623512 0.23363756  19.789251 3.684770e-87  fixed
    #> 4:       alpha -1.088828 0.14275937  -7.627013 2.402557e-14  fixed

and leverage (`rho`)

    summary(leverage, report = "transformed")
    #>    parameter     estimate    std_error   z_value       p_value        type
    #>       <char>        <num>        <num>     <num>         <num>      <char>
    #> 1:   sigma_y  0.008338412 0.0004163314  20.02830  3.121144e-89 transformed
    #> 2:   sigma_h  0.273443559 0.0182641070  14.97164  1.125191e-50 transformed
    #> 3:       phi  0.967721215 0.0043681868 221.53842  0.000000e+00 transformed
    #> 4:       rho -0.748695259 0.0322487815 -23.21623 3.121690e-119 transformed

There is clear evidence for both asymmetric returns and a negative correlation (of -0.74!) between log-returns and the volatility. To find the model that fits the data best, we can compare the AIC of our models and pick the smallest.

    AIC(gaussian,
        t_dist,
        skew_gaussian,
        leverage)
    #>               df       AIC
    #> gaussian       3 -23430.57
    #> t_dist         4 -23451.69
    #> skew_gaussian  4 -23440.87
    #> leverage       4 -23608.85

Clearly the leverage model outperforms the others and is our preferred model for this dataset. Lastly, we can also plot the estimated log-volatility and volatility:

    plot(leverage, include_ci = TRUE, plot_log = TRUE, dates = spy$date)

    plot(leverage, include_ci = TRUE, plot_log = FALSE, dates = spy$date)

We can simulate future returns with `predict`. This function returns three matrices of dimension \(\#\) steps \(\times\) #$ simulations: (1) the latent log-volatility `h`; (2) one for the percentage volatility `100 * sigma_y * exp(0.5 * h)` and (3) future returns. If the argument `include_parameters` is set to `TRUE`, the fixed parameters are simulated from their asymptotic multivariate distribution. This usually leads to broader uncertainty bands. To summarize the output from `predict`, we use the `summary` function, that calculate the mean and different quantiles based on the sumulations. We use the leverage model as an example:

    pred = predict(leverage, steps = 10, include_parameters = TRUE)
    summary(pred)
    #> $y
    #>      time quantile_0.025 quantile_0.975          mean
    #>     <int>          <num>          <num>         <num>
    #>  1:     1    -0.03820274     0.03782478 -1.675297e-04
    #>  2:     2    -0.03643475     0.03708863  1.596740e-04
    #>  3:     3    -0.03646179     0.03654439 -9.517806e-05
    #>  4:     4    -0.03589272     0.03748959  8.243290e-05
    #>  5:     5    -0.03573966     0.03606722  2.457027e-04
    #>  6:     6    -0.03526859     0.03699462  3.736491e-04
    #>  7:     7    -0.03544122     0.03584082 -3.747179e-05
    #>  8:     8    -0.03590485     0.03493844  6.204478e-05
    #>  9:     9    -0.03434944     0.03449313  7.344633e-05
    #> 10:    10    -0.03455827     0.03472973  6.317797e-05
    #>
    #> $h
    #>      time quantile_0.025 quantile_0.975     mean
    #>     <int>          <num>          <num>    <num>
    #>  1:     1     0.36012562       2.489866 1.446109
    #>  2:     2     0.21317695       2.545042 1.398119
    #>  3:     3     0.10260681       2.564030 1.349403
    #>  4:     4    -0.03862903       2.611381 1.306382
    #>  5:     5    -0.12340161       2.650409 1.261903
    #>  6:     6    -0.23088813       2.665538 1.218753
    #>  7:     7    -0.32323770       2.688801 1.176898
    #>  8:     8    -0.42770092       2.673421 1.137958
    #>  9:     9    -0.49593955       2.703493 1.100300
    #> 10:    10    -0.57708990       2.713219 1.065206
    #>
    #> $h_exp
    #>      time quantile_0.025 quantile_0.975       mean
    #>     <int>          <num>          <num>      <num>
    #>  1:     1    0.009949842     0.02896505 0.01780364
    #>  2:     2    0.009198752     0.03003239 0.01755060
    #>  3:     3    0.008676858     0.03069630 0.01724975
    #>  4:     4    0.008176657     0.03098713 0.01698009
    #>  5:     5    0.007869699     0.03153715 0.01675867
    #>  6:     6    0.007375429     0.03173706 0.01639865
    #>  7:     7    0.007068744     0.03207178 0.01617682
    #>  8:     8    0.006712055     0.03187509 0.01597681
    #>  9:     9    0.006427473     0.03248302 0.01569916
    #> 10:    10    0.006144149     0.03269628 0.01554105

    # plot the forecast
    plot(leverage, forecast = 50) + ggplot2::xlim(3200, nrow(spy) + 50)
    #> Warning: Removed 3199 rows containing missing values or values outside the scale range
    #> (`geom_line()`).
    #> Removed 3199 rows containing missing values or values outside the scale range
    #> (`geom_line()`).
    #> Removed 3199 rows containing missing values or values outside the scale range
    #> (`geom_line()`).

# Comparison to stochvol

The R-package `stochvol` (Kastner (2016)) provides a Bayesian framework for inference using Markov Chain Monte Carlo. An advantage of `stochvolTMB` is that optimization can be a lot faster than MCMC. We here compare the leverage and the gaussian model. Depending on your machine you can expect a speed up of 20-200x.

    library(stochvol)

    stochvol_gauss <- svsample(spy$log_return, quiet = T)
    stochvolTMB_gauss  <- estimate_parameters(spy$log_return, "gaussian", silent = TRUE)

    stochvol_lev <- svlsample(spy$log_return, quiet = T)
    stochvolTMB_lev  <- estimate_parameters(spy$log_return, "leverage", silent = TRUE)

We can compare the parameter estimates of the two methods. Note that the parameter `exp(mu/2)` and `sigma` from `stochvol` is the same as `sigma_y` and `sigma_h` from `stochvolTMB`. Both methods give almost identical results.

    stochvol_gauss$para
    #>                   mean           sd          5%          50%          95%
    #> mu        -9.611533785 0.1854455626 -9.91175182 -9.611774497 -9.304678419
    #> phi        0.978189757 0.0049148914  0.96968612  0.978414831  0.985836137
    #> sigma      0.229775055 0.0203786934  0.19733459  0.228942708  0.264098047
    #> exp(mu/2)  0.008217761 0.0007671236  0.00704191  0.008181439  0.009539261
    #> sigma^2    0.053211825 0.0094502446  0.03894094  0.052414764  0.069747778
    #>                 ESS
    #> mu        6230.1953
    #> phi        327.9347
    #> sigma      152.4001
    #> exp(mu/2) 6230.1953
    #> sigma^2    152.4001
    summary(stochvolTMB_gauss, report = "transformed")
    #>    parameter    estimate    std_error   z_value      p_value        type
    #>       <char>       <num>        <num>     <num>        <num>      <char>
    #> 1:   sigma_y 0.008185162 0.0007314791  11.18988 4.570586e-29 transformed
    #> 2:   sigma_h 0.222440223 0.0190056672  11.70389 1.217431e-31 transformed
    #> 3:       phi 0.979034243 0.0046595769 210.11226 0.000000e+00 transformed
    stochvol_lev$para
    #>                   mean           sd           5%          50%          95%
    #> mu        -9.569375399 0.1053851888 -9.739817718 -9.571446103 -9.400002030
    #> phi        0.966952934 0.0041187166  0.960195391  0.966853189  0.973557574
    #> sigma      0.276351629 0.0172514582  0.248100974  0.277044222  0.304057250
    #> rho       -0.724949416 0.0328632332 -0.775144574 -0.726905055 -0.667578807
    #> exp(mu/2)  0.008368349 0.0004417329  0.007674065  0.008348085  0.009095268
    #> sigma^2    0.076667806 0.0095068932  0.061554093  0.076753501  0.092450811
    #>                 ESS
    #> mu        368.69311
    #> phi       140.52279
    #> sigma      42.20112
    #> rho        26.56514
    #> exp(mu/2) 368.69311
    #> sigma^2    42.20112
    summary(stochvolTMB_lev, report = "transformed")
    #>    parameter     estimate    std_error   z_value       p_value        type
    #>       <char>        <num>        <num>     <num>         <num>      <char>
    #> 1:   sigma_y  0.008338412 0.0004163314  20.02830  3.121144e-89 transformed
    #> 2:   sigma_h  0.273443559 0.0182641070  14.97164  1.125191e-50 transformed
    #> 3:       phi  0.967721215 0.0043681868 221.53842  0.000000e+00 transformed
    #> 4:       rho -0.748695259 0.0322487815 -23.21623 3.121690e-119 transformed

# Estimation

The R-package `TMB` (Kristensen et al. (2016)) is used to implement our models for maximum likelihood estimation, since `TMB` lets us estimate parameters in models with a high number of latent variables.

Parameter estimation of stochastic volatility models is hard due to the fact the likelihood function is expressed as a high dimensional integral over the latent variables that cannot be evaluated analytically. If \(\boldsymbol{y} = (y_1, \ldots, y_T)\) denotes our observations, \(\boldsymbol{h} = (h_1, \ldots, h_T)\) our latent variables and \(\boldsymbol{\theta}\) the parameters of interest, the likelihood of \(\boldsymbol{\theta}\) is given by

\[\begin{equation} \mathcal{L}(\boldsymbol{\theta}) = \int f_{\boldsymbol{y}}(\boldsymbol{y}|\boldsymbol{h})f_{\boldsymbol{h}}(\boldsymbol{h}) \, d\boldsymbol{h}, \end{equation}\]

The conditional density of our observations given \(\boldsymbol{h}\) is denoted by \(f_{\boldsymbol{y}}(\boldsymbol{y|u})\), and \(f_{\boldsymbol{h}}(\boldsymbol{h})\) denotes the marginal density of \(\boldsymbol{h}\). To approximate this integral we apply the Laplace approximation.

## Laplace Approximation

Let \(\boldsymbol{y}\) be a vector of observations, \(\boldsymbol{\theta}\) our parameters of interest and \(\boldsymbol{h}\) be a random vector of latent variables. Let \(g(\boldsymbol{h},\boldsymbol{\theta})\) denote the negative joint log-likelihood. The likelihood of \(\boldsymbol{\theta}\) is given by

\[\begin{equation} \mathcal{L}(\boldsymbol{\theta}) = \int f(\boldsymbol{y},\boldsymbol{h}) \, d\boldsymbol{h} = \int f_{\boldsymbol{y}}(\boldsymbol{y|u}) f_{\boldsymbol{h}}(\boldsymbol{h}) \, d\boldsymbol{h} = \int \exp \{-g(\boldsymbol{h},\boldsymbol{\theta})\} \, d\boldsymbol{h}. \end{equation}\]

We assume that \(g\) has a global minimum at \(\boldsymbol{\hat{h}}\) for a given \(\boldsymbol{\theta}\), i.e. \(\boldsymbol{\hat{h}} = \text{argmin}_{\boldsymbol{h}} g(\boldsymbol{h},\boldsymbol{\theta})\), and that \(g\) is twice differentiable. The solution \(\hat{\boldsymbol{h}}\) is known as the Empirical Bayes (EB) estimate. A second order Taylor expansion around \(\boldsymbol{\hat{h}}\) yields

\[\begin{equation} g(\boldsymbol{h},\boldsymbol{\theta}) \approx g(\boldsymbol{\hat{h}},\boldsymbol{\theta}) + \nabla g(\boldsymbol{\hat{h}},\boldsymbol{\theta})(\boldsymbol{h} - \boldsymbol{\hat{h}}) + \frac{1}{2}(\boldsymbol{h} - \boldsymbol{\hat{h}})^T\mathbb{H}(\boldsymbol{h} - \boldsymbol{\hat{h}}) \end{equation}\]

Since \(\boldsymbol{\hat{h}}\) is a minimum, \(\nabla g(\boldsymbol{\hat{h}},\boldsymbol{\theta}) = 0\). Therefore

\[\begin{equation} \mathcal{L}(\boldsymbol{\theta}) \approx \exp \{-g(\boldsymbol{\hat{h}},\boldsymbol{\theta})\} \int \exp \bigg \{-\frac{1}{2}(\boldsymbol{h} - \boldsymbol{\hat{h}})^T\mathbb{H}(\boldsymbol{h} - \boldsymbol{\hat{h}}) \bigg \} d\boldsymbol{h} \end{equation}\]

We can observe that the integrand is the kernel of a multivariate normal density with covariance matrix \(\mathbb{H}^{-1}\). The approximation is therefore given by

\[\begin{equation} \mathcal{L}(\boldsymbol{\theta}) \approx \exp \{-g(\boldsymbol{\hat{h}},\boldsymbol{\theta})\} (2\pi)^{\text{dim}(\boldsymbol{h})/2} \text{det}(\mathbb{H})^{-1/2}, \end{equation}\]

where we have used the fact that \(\text{det}(\mathbb{H}^{-1}) = \text{det}(\mathbb{H})^{-1}\). The corresponding negative log-likelihood is

\[\begin{equation} -l(\boldsymbol{\theta}) = -\frac{\text{dim}(\boldsymbol{h})}{2} \log (2\pi) + \frac{1}{2} \log \text{det}(\mathbb{H}) + g(\boldsymbol{\hat{h}},\boldsymbol{\theta}). \end{equation}\]

Finding the optimal value of \(\boldsymbol{\theta}\) can be viewed as a nested optimization problem. To find \(\boldsymbol{h} (\boldsymbol{\theta})\) and \(\mathbb{H}(\boldsymbol{\theta})\) we fix \(\boldsymbol{\theta}\) and optimize using a quasi-Newton algorithm or a limited memory Newton method. The Laplace approximation is then optimized w.r.t. \(\boldsymbol{\theta}\) using the quasi-Newton algorithm.

# References

Kastner, Gregor. 2016. “Dealing with Stochastic Volatility in Time Series Using the R Package stochvol.” Journal of Statistical Software 69 (5): 1–30. https://doi.org/10.18637/jss.v069.i05.
Kristensen, Kasper, Anders Nielsen, Casper Berg, Hans Skaug, and Bradley Bell. 2016. “TMB: Automatic Differentiation and Laplace Approximation.” Journal of Statistical Software, Articles 70 (5): 1–21. https://doi.org/10.18637/jss.v070.i05.
Taylor, S. J. 1982. “Financial Returns Modelled by the Product of Two Stochastic Processes-a Study of the Daily Sugar Prices 1961-75.” Time Series Analysis : Theory and Practice 1: 203–26. https://cir.nii.ac.jp/crid/1571417125349824256?lang=en.